NTA JEE Mains 23rd Jan 2025 Shift 1 - Mathematics

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be $$\widehat{i}+2\widehat{j}+\widehat{k},\widehat{i}+3\widehat{j}-2\widehat{k}$$ and $$2\widehat{i}+\widehat{j}-\widehat{k}$$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through of the triangle ABC at the point . If the length of AD is $$\frac{\sqrt{110}}{3}$$ and the volume of the tetrahedron is $$\frac{\sqrt{805}}{6\sqrt{2}}$$, then the position vector of is

If A, B and $$(adj (A^{-1})+adj(B^{-1}))$$ are non-singular matrices of same order, then the inverse of $$A(adj(A^{-1}+adj(B^{-1}))^{-1}B$$, is equal to

Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is

Let a curve y=f(x) pass through the points (0,5) and $$(\log_{e}2,k)$$ . If the curve satisfies the differential equation $$2(3+y)e^{2x}dx-(7+e^{2x})dy=0$$ , then k is equal to

If the function $$f(x)=\begin{cases}\frac{2}{x}\{\sin((k_1+1)x)+\sin(k_2-1)x\}, & x<0 \\4, & x=0 \\\frac{2}{x}\log_e\left(\frac{2+k_1x}{2+k_2x}\right), & x>0\end{cases}$$ is continuous at x=0, then $$k_1^2+k_2^2$$ is equal to

If the line 3x-2y+12=0 intersects the parabola $$4y=3x^{2}$$ At the points A and B , then at the vertex of the parabola, the line segment AB subtends an angle equal to

Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line $$\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}$$. Then the area of the right angled triangle PQR , where R is the point (3,-2,1),is

Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that $$\frac{\text{lenght of arc AB}}{\text{lenght of arc BC}}=\frac{1}{5}$$,and $$\overrightarrow{OC}=\alpha\overrightarrow{OA}+\beta\overrightarrow{OB}$$, then $$\alpha +\sqrt{2}(\sqrt{3}-1)\beta$$ is equal to

Let $$f(x)=\log_{e}x$$ and $$g(x)=\frac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$$. Then the domain of $$f \circ g$$ is

If the system of equations $$(\lambda-1)x+(\lambda-4)y+\lambda z=5 \\\lambda x+(\lambda-1)y+(\lambda-4)z=7 \\ (\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$$ has infinitely many solutions, then $$\lambda^{2}+\lambda$$ is equal to

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

Let $$R=\left\{(1,2),(2,3),(3,3)\right\}$$ be a relation defined on the set $$\left\{1,2,3,4\right\}$$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Let the area of a $$\triangle PQR$$ with vertices P(5,4), Q(-2,4) and R(a,b) be 35 square units. If its orthocenter and centroid are $$O(2,\frac{14}{5})$$ and C(c,d) respectively, then c+2d is equal to

The value of $$\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{((log_{e}x)^{2}+1)^{-1}}}{e^{((log_{e}x)^{2}+1)^{-1}}+e^{((6-\log_{e}x)^{2}+1)^{-1}}}\right)dx$$ is

Let $$\mid\frac{\overline{z}-i}{2\overline{z}+i}\mid=\frac{1}{3}, z\in C$$, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and $$(\alpha,0)$$ is 11 square units, then $$\alpha^{2}$$ equals:

The value of $$\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ}\cot 70^{\circ}-1\right)$$ is

Let $$I(x)=\int_{}^{} \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$$. If $$I(37)-I(24)=\frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}}-\frac{1}{c^{\frac{1}{13}}}\right),b,c\in N$$, then 3(b+c) is equal to

If $$\frac{\pi}{2}\leq x\leq \frac{3\pi}{4}$$, then $$\cos^{-1}\left(\frac{12}{13}\cos x+\frac{5}{13}\sin x\right)$$ is equal to

Let the circle touch the line x-y+1=0, have the centre on the positive x -axis, and cut off a chord of length $$\frac{4}{\sqrt{13}}$$ along the line -3x+2y=1. Let H be the hyperbola $$\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$$ , whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then $$2\alpha^{2}+3\beta^{2}$$ is equal to ______

If the equation $$a(b-c)x^{2}+b(c-a)x+c(a-b)=0$$ has equal roots, where a+c=15 and $$b=\frac{36}{5}$$, then $$a^{2}+c^{2}$$ is equal to

If the set of all values of a, for which the equation $$5x^{3}-15x-a=0$$ has three distinct real roots, is the interval $$(\alpha, \beta)$$, then $$\beta-2\alpha $$ is equal to ______

The sum of all rational terms in the expansion of $$(1+2^{1/2}+3^{1/2})^{6}$$ is equal to

If the area of the larger portion bounded between the curves $$x^{2}+y^{2}=25$$ and y=|x-1| is $$\frac{1}{4}(b\pi+c),b,c\in N$$, then b+c is equal to